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1、7TheLogisticRegressionModelInthischapterwewillconsiderregressionmodelswhentheregressandisdichotomousorbinaryinnature.Thedataisoftheform(Y,1X,1)(Y,2X,2)...,(Y,nXn),wherethedependentvariableY,i=1,…,naretheobservedbinaryoutputsiassumedtobeindependent(inthestatisticalsense)ofeachother,the
2、vectorX,ii=1,…,n,arethecovariates(independentvariablesinthesenseofaregressionproblem)associatedwithYi.Inthepreviouschapterweconsideredthelinearregressionmodelwheretheregressandwasassumedtobecontinuousalongwiththeassumptionofnormalityfortheerrordistribution.Here,wewillconsideraGaussian
3、(normal)modelforthebinaryregressionmodel,whichismorewidelyknownastheprobitmodel.Amoregenericmodelhasemergedduringthepastfourdecadesintheformofalogisticregressionmodel.Wewillconsiderthelogisticregressionmodelforrestofthechapter.Theapproachinthischapterwillbeonthefollowinglines:Thebinar
4、yregressionproblemTheprobitregressionmodelThelogisticregressionmodelModelvalidationanddiagnosticsReceivingoperatorcurvesLogisticregressionfortheGermancreditscreeningdatasetTheLogisticRegressionModelThebinaryregressionproblemConsidertheproblemofmodelingthecompletionofastatcoursebystude
5、ntsbasedontheirScholasticAssessmentTestinthesubjectofmathematicsSAT-Mscoresatthetimeoftheiradmission.Afterthecompletionofthefinalexamsweknowwhichstudentssuccessfullycompletedthecourseandwhichofthemfailed.Here,theoutputpass/failmayberepresentedbyabinarynumber1/0.Itmaybefairlysaidthathi
6、ghertheSAT-Mscoresatthetimeofadmissiontothecourse,themorelikelihoodofthecandidatecompletingthecourse.ThisproblemhasbeendiscussedindetailinJohnsonandAlbert(1999)andTattar,et.al.(2013).Letusbeginbydenotingthepass/failindicatorbyYandtheentrySAT-MscorebyX.Supposethatwehavenpairsofobservat
7、ionsonthestudents'scoresandtheircoursecompletionresults.Wecanbuildthesimplelinearregressionmodelfortheprobabilityofcoursecompletionpi=P(Yi=1)asafunctionoftheSAT-Mscorewithpi=+ββ00Xi+εi.Thedatafrompage77ofJohnsonandAlbert(1999)isavailableinthesatdatasetofthebook'sRpackage.Thecolumnstha
8、tcont
